Simplify the following expressions: (42)4 / 48, aba2b5, a5ⁿ, and (13/10)3.

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Simplify the expression using exponent rules.
For , we use the power of a power rule $(x^m)^n = x^{mn}$ and the quotient rule $\frac{x^m}{x^n} = x^{m-n}$ .
$\frac{(4^2)^4}{4^8} = \frac{4^{2 \times 4}}{4^8}$ $= \frac{4^8}{4^8}$ $= 4^{8-8}$ $= 4^0$ $= 1$
The simplified expression is $\boxed{1}$ .
Simplify the expression by combining like terms.
For , we use the product rule $x^m \times x^n = x^{m+n}$ . Remember that $a = a^1$ and $b = b^1$ .
$a \times b \times a^2 \times b^5 = (a^1 \times a^2) \times (b^1 \times b^5)$ $= a^{1+2} \times b^{1+5}$ $= a^3 \times b^6$ $= a^3 b^6$
The simplified expression is $\boxed{a^3 b^6}$ .
Evaluate the expression.
For , the expression $a \times 5^n$ is already in its simplest form. There are no further operations to perform without specific values for $a$ or $n$ .
The expression is $\boxed{a \times 5^n}$ .
Evaluate the power of a fraction.
For , we apply the power to both the numerator and the denominator: $(\frac{x}{y})^n = \frac{x^n}{y^n}$ .
$\left(\frac{13}{10}\right)^3 = \frac{13^3}{10^3}$
Calculate the powers:
$13^3 = 13 \times 13 \times 13 = 169 \times 13 = 2197$ $10^3 = 10 \times 10 \times 10 = 1000$
So,
$\left(\frac{13}{10}\right)^3 = \frac{2197}{1000}$
The evaluated expression is $\boxed{\frac{2197}{1000}}$ .

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